Real, who began the game bottom of Group A, clinched their first win in the tournament this season to ease the pressure on boss Zinedine Zidane – Jose Mourinho has been linked with his job.
Zidane's side were knocked off top spot in LaLiga by Barcelona after Saturday's 1-0 defeat at Mallorca but responded with a crucial win, which could have been by a wider margin had it not been for Galatasaray goalkeeper Fernando Muslera.
Muslera made a string of outstanding saves to keep Galatasaray in the tie, while the Turkish side were not without their chances.
Andone forced Courtois into a better save, low in the bottom corner, but after efforts from Rodrygo and Karim Benzema had both tested Muslera, Real took an 18th-minute lead.
Eden Hazard cut the ball back inside the area and Kroos marked his 100th Champions League appearance with a deflected shot, which flew into the top corner and silenced the partisan home fans.
Andone almost provided a speedy response when he headed Yuto Nagatomo's cross narrowly off target five minutes later.
Muslera kept out Hazard's shot, but Real were not having it all their own way as Belhanda tested Courtois again before Federico Valverde sizzling drive just before half-time was superbly turned away by the Turkish champions' goalkeeper.
Kroos sent Benzema clear early in the second period and the Frenchman's rising drive was saved at full stretch by Muslera.
Andone had header and shot both blocked and Steven Nzonzi blazed high and wide following a corner as Galatasaray looked to get back on level terms.
But it was Muslera who they had to thank for trailing by just the one goal as he saved efforts from Hazard and Dani Carvajal before the Belgian crashed a shot against the crossbar after being put through by Benzema.
Rodrygo and Benzema were next to be denied by Muslera before Nzonzi spurned a golden chance to equalise, heading wide from six yards.
Muslera pulled off further saves to thwart Vinicius Junior and Casemiro either side of Christian Luyindama's blocked header for the home side in a frantic finish.